Let
be a set of random variables on
.
A coherent measure is a mapping:
:

(Coherent measure)

Proposition

The coherent measure has the following properties:

Proposition

Proposition

1. (Subadditivity)
.

2. (Monotonicity)
.

3. (Positive homogeneity)
for
.

4. (Translation invariance).
.

5. (Fatou property).
in
.

Definition

Given a coherent measure
,
the largest set
that enables the relationship (
Coherent
measure
) is called the determining set
.

Note that if
is finite then the representation (
Coherent
measure
) is exactly the formula (
Support
function
). Hence, for the finite
the representation (
Coherent measure
)
completely describes all functions that have the property 1 and 3.
Furthermore, if we introduce a notion of maximal set
(in inclusion sense) for a measure
then, given
and
as in (
Coherent measure
), taking double
convex dual of
reveals that the
is a closure of the convex hull of the original
.
Another description for
is
where the
is the set of all absolutely continuous measures with respect to
.

In the case of general
there is a following representation theorem.

Proposition

A measure
satisfies the conditions 1-5 above iff there exists a non empty
such that the (
Coherent measure
) holds.

Note that the theorem holds for
.
There reason may be easily understood by looking at derivations around the
formula (
Support function
). To extend those
derivations to the general case we need the duality to the space
in order to make sense of all the scalar products
.
Such duality needs to have reflexivity. The
has such dual space
while
,
endowed with the topology of convergence in probability, does not have such
structure.